Questions: Question 6 of 16 (1 point) Question Attempt 1 of Unlimited Fifteen items or less: The number of customers in line at a supermarket express checkout counter is a random variable with the following probability distribution. x 0 1 2 3 4 5 P(x) 0.05 0.25 0.30 0.20 0.15 0.05 Part 1 of 4 (a) Find P(3). P(3)=

Solution Steps

The probability of having 3 customers in line at the supermarket express checkout counter is given by:

\[ P(3) = 0.20 \]

The mean (expected value) of the distribution is calculated as follows:

\[ \text{Mean} = E(X) = \sum_{x=0}^{5} x \cdot P(x) = 0 \times 0.05 + 1 \times 0.25 + 2 \times 0.30 + 3 \times 0.20 + 4 \times 0.15 + 5 \times 0.05 = 2.3 \]

The variance \( \sigma^2 \) is calculated using the formula:

\[ \text{Variance} = \sigma^2 = \sum_{x=0}^{5} (x - \text{Mean})^2 \cdot P(x) = (0 - 2.3)^2 \times 0.05 + (1 - 2.3)^2 \times 0.25 + (2 - 2.3)^2 \times 0.3 + (3 - 2.3)^2 \times 0.2 + (4 - 2.3)^2 \times 0.15 + (5 - 2.3)^2 \times 0.05 = 1.61 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \text{Standard Deviation} = \sigma = \sqrt{\sigma^2} = \sqrt{1.61} \approx 1.269 \]

Final Answer